A sum of subharmonic and superharmonic function that is subharmonic
$begingroup$
Suppose $u$ and $v$ are subharmonic on a bounded domain $G$ of $mathbb{R}^{n}$ with $ngeq2$, and $w=u-v.$ If $w$ is also subharmonic and defined everywhere on $G$ (we exclude the case $u$ or $v$ is harmonic), can we say that $u$ and $v$ differ by a constant?
real-analysis complex-analysis probability-theory potential-theory
asked Dec 25 '18 at 3:17
M. RahmatM. Rahmat
306213
$endgroup$
add a comment |
$begingroup$
Suppose $u$ and $v$ are subharmonic on a bounded domain $G$ of $mathbb{R}^{n}$ with $ngeq2$, and $w=u-v.$ If $w$ is also subharmonic and defined everywhere on $G$ (we exclude the case $u$ or $v$ is harmonic), can we say that $u$ and $v$ differ by a constant?
real-analysis complex-analysis probability-theory potential-theory
asked Dec 25 '18 at 3:17
M. RahmatM. Rahmat
306213
$endgroup$
add a comment |
$begingroup$
Suppose $u$ and $v$ are subharmonic on a bounded domain $G$ of $mathbb{R}^{n}$ with $ngeq2$, and $w=u-v.$ If $w$ is also subharmonic and defined everywhere on $G$ (we exclude the case $u$ or $v$ is harmonic), can we say that $u$ and $v$ differ by a constant?
real-analysis complex-analysis probability-theory potential-theory
asked Dec 25 '18 at 3:17
M. RahmatM. Rahmat
306213
$endgroup$
Suppose $u$ and $v$ are subharmonic on a bounded domain $G$ of $mathbb{R}^{n}$ with $ngeq2$, and $w=u-v.$ If $w$ is also subharmonic and defined everywhere on $G$ (we exclude the case $u$ or $v$ is harmonic), can we say that $u$ and $v$ differ by a constant?
real-analysis complex-analysis probability-theory potential-theory
real-analysis complex-analysis probability-theory potential-theory
asked Dec 25 '18 at 3:17
M. RahmatM. Rahmat
306213
asked Dec 25 '18 at 3:17
M. RahmatM. Rahmat
306213
edited Dec 25 '18 at 6:14
M. Rahmat
asked Dec 25 '18 at 3:17
M. RahmatM. Rahmat
306213
asked Dec 25 '18 at 3:17
M. RahmatM. Rahmat
306213
asked Dec 25 '18 at 3:17
M. RahmatM. Rahmat
306213
306213
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I don't know if I have misunderstood the question but if $v,w$ are (finite) subharmonic and $u=v+w$ then the hypothesis is satisfied, right?
answered Dec 25 '18 at 5:33
Kavi Rama MurthyKavi Rama Murthy
77.9k53471
$endgroup$
$begingroup$
Thanks for noticing it. You are right! I edited and excluded this case.
$endgroup$
– M. Rahmat
Dec 25 '18 at 6:15
1
$begingroup$
@M.Rahmat That doesn't make any difference. Sum of two subharmonic functions is subharmonic . I have changed harmonic to subharmonic in my answer.
$endgroup$
– Kavi Rama Murthy
Dec 25 '18 at 6:19
add a comment |
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051811%2fa-sum-of-subharmonic-and-superharmonic-function-that-is-subharmonic%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I don't know if I have misunderstood the question but if $v,w$ are (finite) subharmonic and $u=v+w$ then the hypothesis is satisfied, right?
answered Dec 25 '18 at 5:33
Kavi Rama MurthyKavi Rama Murthy
77.9k53471
$endgroup$
$begingroup$
Thanks for noticing it. You are right! I edited and excluded this case.
$endgroup$
– M. Rahmat
Dec 25 '18 at 6:15
1
$begingroup$
@M.Rahmat That doesn't make any difference. Sum of two subharmonic functions is subharmonic . I have changed harmonic to subharmonic in my answer.
$endgroup$
– Kavi Rama Murthy
Dec 25 '18 at 6:19
add a comment |
$begingroup$
I don't know if I have misunderstood the question but if $v,w$ are (finite) subharmonic and $u=v+w$ then the hypothesis is satisfied, right?
answered Dec 25 '18 at 5:33
Kavi Rama MurthyKavi Rama Murthy
77.9k53471
$endgroup$
$begingroup$
Thanks for noticing it. You are right! I edited and excluded this case.
$endgroup$
– M. Rahmat
Dec 25 '18 at 6:15
1
$begingroup$
@M.Rahmat That doesn't make any difference. Sum of two subharmonic functions is subharmonic . I have changed harmonic to subharmonic in my answer.
$endgroup$
– Kavi Rama Murthy
Dec 25 '18 at 6:19
add a comment |
$begingroup$
I don't know if I have misunderstood the question but if $v,w$ are (finite) subharmonic and $u=v+w$ then the hypothesis is satisfied, right?
answered Dec 25 '18 at 5:33
Kavi Rama MurthyKavi Rama Murthy
77.9k53471
$endgroup$
I don't know if I have misunderstood the question but if $v,w$ are (finite) subharmonic and $u=v+w$ then the hypothesis is satisfied, right?
answered Dec 25 '18 at 5:33
Kavi Rama MurthyKavi Rama Murthy
77.9k53471
edited Dec 25 '18 at 6:18
answered Dec 25 '18 at 5:33
Kavi Rama MurthyKavi Rama Murthy
77.9k53471
answered Dec 25 '18 at 5:33
Kavi Rama MurthyKavi Rama Murthy
77.9k53471
answered Dec 25 '18 at 5:33
Kavi Rama MurthyKavi Rama Murthy
77.9k53471
77.9k53471
$begingroup$
Thanks for noticing it. You are right! I edited and excluded this case.
$endgroup$
– M. Rahmat
Dec 25 '18 at 6:15
1
$begingroup$
@M.Rahmat That doesn't make any difference. Sum of two subharmonic functions is subharmonic . I have changed harmonic to subharmonic in my answer.
$endgroup$
– Kavi Rama Murthy
Dec 25 '18 at 6:19
add a comment |
$begingroup$
Thanks for noticing it. You are right! I edited and excluded this case.
$endgroup$
– M. Rahmat
Dec 25 '18 at 6:15
1
$begingroup$
@M.Rahmat That doesn't make any difference. Sum of two subharmonic functions is subharmonic . I have changed harmonic to subharmonic in my answer.
$endgroup$
– Kavi Rama Murthy
Dec 25 '18 at 6:19
$begingroup$
Thanks for noticing it. You are right! I edited and excluded this case.
$endgroup$
– M. Rahmat
Dec 25 '18 at 6:15
$begingroup$
Thanks for noticing it. You are right! I edited and excluded this case.
$endgroup$
– M. Rahmat
Dec 25 '18 at 6:15
1
1
$begingroup$
@M.Rahmat That doesn't make any difference. Sum of two subharmonic functions is subharmonic . I have changed harmonic to subharmonic in my answer.
$endgroup$
– Kavi Rama Murthy
Dec 25 '18 at 6:19
$begingroup$
@M.Rahmat That doesn't make any difference. Sum of two subharmonic functions is subharmonic . I have changed harmonic to subharmonic in my answer.
$endgroup$
– Kavi Rama Murthy
Dec 25 '18 at 6:19
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051811%2fa-sum-of-subharmonic-and-superharmonic-function-that-is-subharmonic%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
